1. Field of the Invention
The present invention relates to a method and apparatus for configuring a quantum mechanical state required for quantum information processing, quantum communication, or quantum precision measurements, as well as a communication method and apparatus using such a method and apparatus.
2. Related Background Art
The fields of quantum computations (see R. P. Feynman, xe2x80x9cFeynman Lectures on Computation,xe2x80x9d Addison-Wesley (1996)) and quantum information theories are advancing rapidly. In these fields, superposition, interference, and an entangled state, which are the basic nature of quantum mechanics, are ingeniously utilized.
In the field of quantum computations, since the publication of Shor""s algorithm concerning factorization (see P. W. Shor, xe2x80x9cPolynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,xe2x80x9d LANL quantum physics archive quant-ph/9508027. Similar contents are found in SIAM J. Computing 26 (1997), 1484. In addition, the first document is P. W. Shor, xe2x80x9cAlgorithms for quantum computation: Discrete logarithms and factoring,xe2x80x9d in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (ed. S. Goldwasser) 124-134 (IEEE Computer Society, Los Alamitos, Calif., 1994). A detailed description is found in Artur Ekert and Richard Jozsa, xe2x80x9cQuantum computation and Shor""s factoring algorithm,xe2x80x9d Rev. Mod. Phys. 68, 733 (1996)) and Grover""s algorithm concerning the search problem (L. K. Grover, xe2x80x9cA fast quantum mechanical algorithm for database search,xe2x80x9d LANL quantum physics archive quant-ph/9605043. Almost similar contents are found in L. K. Grover, xe2x80x9cQuantum Mechanics Helps in Searching for a Needle in a Haystack,xe2x80x9d Phys. Rev. Lett. 79, 325 (1997)), many researchers have been proposing methods for implementing quantum computations and developing new quantum algorithms.
On the other hand, in the field of quantum information theories, the entangled state has been known to play an important role due to its unlikelihood to be affected by decoherence (see C. H. Bennett, C. A. Fuchs, and J. A. Smolin, xe2x80x9cEntanglement-Enhanced Classical Communication on a Noisy Quantum Channel,xe2x80x9d Quantum Communication, Computing, and Measurement, edited by Hirota et al., Plenum Press, New York, p. 79 (1997)).
Furthermore, as an application of these results, a method for overcoming the quantum shot noise limit using (n) two-level entangled states has been established through experiments on Ramsey spectroscopy (see D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heizen, xe2x80x9cSpin squeezing and reduced quantum noise in spectroscopy,xe2x80x9d Phys. Rev. A 46}, R6797 (1992) or D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heizen, xe2x80x9cSqueezed atomic states and projection noise in spectroscopy,xe2x80x9d Phys. Rev. A 50, 67 (1994)). Despite the lack of discussion of the two-level Ramsey spectroscopy, a similar concept is described in M. Kitagawa and M. Ueda, xe2x80x9cNonlinear-Interferometric Generation of Number-Phase-Correlated Fermion States,xe2x80x9d Phys. Rev. Lett. 67, 1852 (1991).
If decoherence in the system caused by the environment is negligible, the maximally entangled state serves to improve the accuracy in measuring the frequency of an energy spectrum.
In this case, the fluctuation of the frequency decreases by 1/{square root over (n)}. With decoherence in the system considered, however, the resolution achieved by the maximally entangled state is only equivalent to that achieved by an uncorrelated system.
In addition, the use of a partly entangled state having a high symmetry has been proposed in S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, J. I. Cirac, xe2x80x9cImprovement of Frequency Standards with Quantum Entanglement,xe2x80x9d Phys. Rev. Lett. 79, 3865 (1997).
If optimal parameters (coefficients of basic vectors) can be selected beforehand, this method can provide a higher resolution than the maximally entangled or uncorrelated state.
The partly entangled state having a high symmetry is given by the following equation:                                           "LeftBracketingBar"                          ψ              n                                ⟩                =                                            ∑                              k                =                0                                            ⌊                                  n                  /                  2                                ⌋                                      ⁢                          xe2x80x83                        ⁢                                          a                k                            ⁢                                                (                                                            "LeftBracketingBar"                      k                                        ⟩                                    )                                s                            ⁢                              xe2x80x83                            ⁢              for              ⁢                              xe2x80x83                            ⁢              n                                ≧          2                                    (        1        )            
where (n) represents the number of qubits that are two-level particles constituting a state, and └n/2┘ represents a maximum integer not more than n/2. {ak} is a real number wherein, for example, an optimal combination of values are assumed to be provided beforehand so as to provide a high resolution in the Ramsey spectroscopy. In this case, {ak} may be a constant.
|k greater than 5 is a superposition of states in which (k) or (nxe2x88x92k) qubits are excited, wherein the superposition is established using an equal weight. For example, |"psgr"4 greater than  is given by the following equation.                                                                                                                                     "LeftBracketingBar"                                              ψ                        4                                                              ⟩                                    =                                      xe2x80x83                                    ⁢                                                            a                      0                                        ⁢                                          "LeftBracketingBar"                      0                                                                      ⟩                            +                                                a                  1                                ⁢                                  "LeftBracketingBar"                  1                                                      ⟩                    +                                    a              2                        ⁢                          "LeftBracketingBar"              2                                      ⟩                                =                  xe2x80x83                ⁢                                                                                                                        a                      0                                        (                                          "LeftBracketingBar"                      0000                                                        ⟩                                +                                  "LeftBracketingBar"                  1111                                            ⟩                        )                    +                                                  xe2x80x83                ⁢                              a            1                    (                                                                                                                                                                                                                                                                                                                                  "LeftBracketingBar"                                    0001                                                                    ⟩                                                                +                                                                  "LeftBracketingBar"                                  0010                                                                                            ⟩                                                        +                                                          "LeftBracketingBar"                              0100                                                                                ⟩                                                +                                                  "LeftBracketingBar"                          1000                                                                    ⟩                                        +                                          "LeftBracketingBar"                      1110                                                        ⟩                                +                                  "LeftBracketingBar"                  1101                                            ⟩                        +                                                                        xe2x80x83                    ⁢                                                                                          "LeftBracketingBar"                    1011                                    ⟩                                +                                  "LeftBracketingBar"                  0111                                            ⟩                        )                          +                              a            2                    (                                                                                                                                                                                              "LeftBracketingBar"                            0011                                                    ⟩                                                +                                                  "LeftBracketingBar"                          0101                                                                    ⟩                                        +                                          "LeftBracketingBar"                      0110                                                        ⟩                                +                                  "LeftBracketingBar"                  1001                                            ⟩                        +                                                            xe2x80x83                ⁢                                                                              "LeftBracketingBar"                  1010                                ⟩                            +                              "LeftBracketingBar"                1100                                      ⟩                    )                    
These states have symmetry such as that described below.
Invariable despite the substitution of any two qubits
Invariable despite the simultaneous inversion of {|0 greater than , |1 greater than } for each qubit.
To conduct experiments on the Ramsey spectroscopy using a partly entangled state having a high symmetry, a target entangled state must be provided as soon as possible as an initial system state before decoherence may occur. Thus, all actual physical systems have a decoherence time, and quantum mechanical operations must be performed within this time. This is a problem in not only quantum precision measurements but also quantum communication and general quantum computations.
In addition, to configure the target entangled state from a specified state, an operation using basic quantum gates must be performed out many times. Minimizing this number of times leads to the reduction of the time required to provide the target entangled state.
In addition, to configure a quantum gate network for obtaining the target entangled state, a conventional computer must be used to determine in advance which qubits will be controlled by the basic quantum gates, the order in which the basic quantum gates will be used, and a rotation parameter for unitary rotations. The amount of these computations is desirably reduced down to an actually feasible level.
Thus, an object of the present invention is to provide a method used to configure a target entangled state within a decoherence time in an actual physical system, for configuring the desired state using as less steps as possible if the computation time is evaluated based on the total number of basic quantum gates.
Another object of the present invention is to provide an effective procedure used in configuring a network consisting of basic quantum gates, for using a conventional computer to determine in advance which be qubits will be controlled by the basic quantum gates, the order in which the basic quantum gates will be used, and a unitary rotation parameter.
Yet another object of the present invention is to provide a method and apparatus for configuring not only a partly entangled state having a high symmetry but also a partly entangled state defined by a function with an even number of collisions, using as less steps as possible if the computation time is evaluated based on the total number of basic quantum gates.
According to one aspect, the present invention which achieves these objects relates to a method for configuring a quantum mechanical state consisting of a plurality of two-level systems wherein if a superposition of orthonormal bases in which each two-level system assumes a basic or an excited state is used for expression, a desired partly-entangled quantum mechanical state in which the coefficients of the bases are all real numbers is configured using an operation that is a combination of a selective rotation operation and an inversion about average operation.
According to one aspect, the present invention which achieves these objectives relates to a state configuration apparatus comprising a selective rotation operation means for a plurality of two-level systems and an inversion about average operation means for the plurality of two-level systems, wherein if a quantum mechanical state consisting of the plurality of two-level systems is expressed by a superposition of orthonormal bases in which each two-level system assumes a basic or an excited state, a desired partly-entangled quantum mechanical state in which the coefficients of the bases are all real numbers is configured using an operation that is a combination of an operation performed by the selective rotation operation means and an operation performed by the inversion about average operation means.
Other objectives and advantages besides those discussed above shall be apparent to those skilled in the art from the description of a preferred embodiment of the invention which follows. In the description, reference is made to accompanying drawings, which form a part thereof, and which illustrate an example of the invention. Such an example, however, is not exhaustive of the various embodiments of the invention, and therefore reference is made to the claims which follow the description for determining the scope of the invention.